Physica D: Nonlinear Phenomena [SJR: 1.048] [H-I: 89] [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0167-2789 Published by Elsevier [2799 journals] |
- Stochastic stability of measures in gradient systems
- Abstract: Publication date: Available online 9 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Wen Huang, Min Ji, Zhenxin Liu, Yingfei Yi
Stochastic stability of a compact invariant set of a finite dimensional, dissipative system is studied in our recent work “Concentration and limit behaviors of stationary measures” (Huang et al., 2015) for general white noise perturbations. In particular, it is shown under some Lyapunov conditions that the global attractor of the systems is always stable under general noise perturbations and any strong local attractor in it can be stabilized by a particular family of noise perturbations. Nevertheless, not much is known about the stochastic stability of an invariant measure in such a system. In this paper, we will study the issue of stochastic stability of invariant measures with respect to a finite dimensional, dissipative gradient system with potential function f . As we will show, a special property of such a system is that it is the set of equilibria which is stable under general noise perturbations and the set S f of global minimal points of f which is stable under additive noise perturbations. For stochastic stability of invariant measures in such a system, we will characterize two cases of f , one corresponding to the case of finite S f and the other one corresponding to the case when S f is of positive Lebesgue measure, such that either some combined Dirac measures or the normalized Lebesgue measure on S f is stable under additive noise perturbations. However, we will show by constructing an example that such measure stability can fail even in the simplest situation, i.e., in 1 -dimension there exists a potential function f such that S f consists of merely two points but no invariant measure of the corresponding gradient system is stable under additive noise perturbations. Crucial roles played by multiplicative and additive noise perturbations to the measure stability of a gradient system will also be discussed. In particular, the nature of instabilities of the normalized Lebesgue measure on S f under multiplicative noise perturbations will be exhibited by an example.
PubDate: 2015-10-11T17:52:05Z
- Abstract: Publication date: Available online 9 October 2015
- Radial symmetry on three-dimensional shells in the Landau-de Gennes theory
- Abstract: Publication date: Available online 9 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Giacomo Canevari, Mythily Ramaswamy, Apala Majumdar
We study the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau-de Gennes theory for nematic liquid crystals. We prove that the radial-hedgehog solution is the unique minimizer of the Landau-de Gennes energy in two separate regimes: (i) for thin shells when the temperature is below the critical nematic supercooling temperature and (ii) for a fixed shell width at sufficiently low temperatures. In case (i), we provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution.
PubDate: 2015-10-11T17:52:05Z
- Abstract: Publication date: Available online 9 October 2015
- Stratification and enumeration of Boolean functions by canalizing depth
- Abstract: Publication date: Available online 8 October 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Qijun He, Matthew Macauley
Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively “picked off”, until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, we obtain closed formulas for the number of n -variable Boolean functions with depth k , which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions.
PubDate: 2015-10-11T17:52:05Z
- Abstract: Publication date: Available online 8 October 2015
- Large-scale weakly nonlinear perturbations of convective magnetic dynamos
in a rotating layer- Abstract: Publication date: Available online 30 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): R. Chertovskih, V. Zheligovsky
We present a new mechanism for generation of large-scale magnetic field by thermal convection which does not involve the α -effect. We consider weakly nonlinear perturbations of space-periodic steady convective magnetic dynamos in a rotating layer of incompressible electrically conducting fluid that were identified in our previous work. The perturbations have a spatial scale in the horizontal direction that is much larger than the period of the perturbed convective magnetohydrodynamic state. Following the formalism of the multiscale stability theory, we have derived the system of amplitude equations governing the evolution of the leading terms in the expansion of the perturbations in power series in the scale ratio. This asymptotic analysis is more involved than in the cases considered earlier, because the kernel of the operator of linearisation has zero-mean neutral modes whose origin lies in the spatial invariance of the perturbed regime, the operator reduced on the generalised kernel has two Jordan normal form blocks of size two, and simplifying symmetries of the perturbed state are now missing. Numerical results for the amplitude equations show that a large-scale perturbation, periodic in slow horizontal variable, either converges to a short-scale neutral stability mode with amplitudes tending to constant values, or it blows up at a finite slow time.
PubDate: 2015-10-03T18:52:10Z
- Abstract: Publication date: Available online 30 September 2015
- Membrane tension feedback on shape and motility of eukaryotic cells
- Abstract: Publication date: Available online 25 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Benjamin Winkler, Igor S. Aranson, Falko Ziebert
In the framework of a phase field model of a single cell crawling on a substrate, we investigate how the properties of the cell membrane affect the shape and motility of the cell. Since the membrane influences the cell dynamics on multiple levels and provides a nontrivial feedback, we consider the following fundamental interactions: (i) the reduction of the actin polymerization rate by membrane tension; (ii) area conservation of the cell’s two-dimensional cross-section vs. conservation of the circumference (i.e. membrane inextensibility); and (iii) the contribution from the membrane’s bending energy to the shape and integrity of the cell. As in experiments, we investigate two pertinent observables—the cell’s velocity and its aspect ratio. We find that the most important effect is the feedback of membrane tension on the actin polymerization. Bending rigidity has only minor effects, visible mostly in dynamic reshaping events, as exemplified by collisions of the cell with an obstacle.
PubDate: 2015-09-25T18:33:46Z
- Abstract: Publication date: Available online 25 September 2015
- Asymptotic analysis of a viscous thread extending under gravity
- Abstract: Publication date: Available online 21 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Jonathan J. Wylie, Huaxiong Huang, Robert M. Miura
Despite extensive research on extensional flows, there is no complete explanation for why highly viscous fluids extending under gravity can form such persistent and stable filaments with no sign of destabilization from surface tension. We therefore investigate the motion of a slender axisymmetric viscous thread that is supported at its top by a fixed horizontal surface and extends downward under gravity. In the case in which inertia and surface tension are initially negligible, we consider the long-wavelength equations for the full initial-boundary-value problem for a thread of arbitrary initial shape. We show that, eventually, the accelerations in the thread become sufficiently large that the inertial terms become important. Thus, we keep the inertial terms and, using matched asymptotic expansions, obtain solutions for the full initial-boundary-value problem. We show that the dynamics can be divided into two generic cases that exhibit very different behaviour. In the first case, the thread develops a long thin region that joins together two fluid masses. In this case, we use order-of-magnitude estimates to show that surface-tension-driven pinching will not occur if the square root of the Reynolds number is much greater than the initial aspect ratio divided by the Bond number. In the second case, the thread becomes thin near the horizontal surface. In this case, we show that the long-wavelength equations will ultimately break down and discuss the role of inertia in determining the dynamics. The asymptotic procedures require a number of novel techniques and the resulting solutions exhibit surprisingly rich behavior. The solution allows us to understand the mechanisms that underlie highly persistent filaments.
PubDate: 2015-09-25T18:33:46Z
- Abstract: Publication date: Available online 21 September 2015
- The effects of wind and nonlinear damping on rogue waves and permanent
downshift- Abstract: Publication date: Available online 21 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): C.M. Schober, M. Strawn
In this paper we investigate the effects of wind and nonlinear damping on permanent downshift and the formation of rogue waves in the framework of a HONLS model. Wind effects are incorporated by including a uniform linear damping/forcing term in the model. The strength of the wind, Γ , is allowed to vary as well as wind duration. Determining permanent downshift is not straightforward and we propose a criteria for permanent downshift related to our numerical experiments. We consider large ensembles of initial data for modulated unstable Stokes waves with N = 1 , 2 , 3 unstable modes. In the nonlinear damped HONLS evolution we find permanent downshift is observed whenever the strength of the nonlinear damping β > 0.1 . Notably, rogue waves typically do not develop after the time of permanent downshift, implying that a downshifted sea-state does not allow for any further rogue waves. Incorporating wind effects into the nonlinear damped HONLS model, we find that damping by the wind weakens downshifting while forcing by the wind enhances downshifting. The proximity of the initial data to unstable plane waves impacts the characteristic features of the rogue waves in the nonlinear damped HONLS evolution. We find that as the initial data is chosen closer to the plane wave, the maximum strength, number, and lifetime of rogue waves increase while the time of permanent downshift decreases. Alternatively, we show that the greater the wave strength, the more rogue waves, or the longer their lifetime, the earlier permanent downshift occurs.
PubDate: 2015-09-25T18:33:46Z
- Abstract: Publication date: Available online 21 September 2015
- Exact solutions of the Hirota equation and vortex filaments motion
- Abstract: Publication date: Available online 21 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): F. Demontis, G. Ortenzi, C. van der Mee
By using the Inverse Scattering Transform we construct an explicit soliton solution formula for the Hirota equation. The formula obtained allows one to get, as a particular case, the N -soliton solution, the breather solution and, most relevantly, a new class of solutions called multipole soliton solutions. We use these exact solutions to study the motion of a vortex filament in an incompressible Euler fluid with nonzero axial velocity.
PubDate: 2015-09-25T18:33:46Z
- Abstract: Publication date: Available online 21 September 2015
- Editorial Board
- Abstract: Publication date: 15 August 2015
Source:Physica D: Nonlinear Phenomena, Volume 310
PubDate: 2015-09-21T18:26:56Z
- Abstract: Publication date: 15 August 2015
- Singularity confinement and full-deautonomisation: A discrete
integrability criterion- Abstract: Publication date: Available online 14 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): B. Grammaticos, A. Ramani, R. Willox, T. Mase, J. Satsuma
We present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector. Our method is based on the full-deautonomisation procedure, which consists in analysing non-autonomous extensions of a given discrete system obtained by adding terms that are initially absent, but whose presence does not alter the singularity pattern. A justification for this approach is given through an algebro-geometric analysis. We also introduce the notions of early and late confinement. While the former is a confinement that may exist already for the autonomous system, the latter corresponds to a singularity pattern longer than that of the autonomous case. Late confinement will be shown to play an important role in the singularity analysis of systems with non-trivial gauge freedom, for which the existence of an undetected gauge in conjunction with a sketchy analysis, might lead to erroneous conclusions as to their integrability. An algebro-geometric analysis of the role of late confinement in this context is also offered. This novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping.
PubDate: 2015-09-18T07:48:29Z
- Abstract: Publication date: Available online 14 September 2015
- Numerical analysis of the subcritical feature of electro-thermo-convection
in a plane layer of dielectric liquid- Abstract: Publication date: Available online 11 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Jian Wu, Philippe Traoré, Alberto T. Pérez, Mengqi Zhang
This paper reports a numerical investigation with a horizontal layer of dielectric liquid subjected to the simultaneous effects of external thermal and electric fields. The flow is driven by the buoyancy force and the Coulomb force, which depend on the Rayleigh number (Ra) and the electric Rayleigh number ( T ), respectively. We consider a strong unipolar injection from the lower electrode and a destabilizing thermal gradient. The two driving forces cooperate with each other in destabilizing the system. The neutral stability curve in the Ra– T plane is successfully reproduced from the direct numerical results, and it is shown to be independent on the Prandtl number (Pr) and the dimensionless mobility number ( M ). On the other hand, both the bifurcation types of the linear instability (subcritical or supercritical) and the finite amplitude stability criterion depend strongly on the combination of Pr and M .
PubDate: 2015-09-14T07:41:12Z
- Abstract: Publication date: Available online 11 September 2015
- Predictability of threshold exceedances in dynamical systems
- Abstract: Publication date: Available online 10 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Tamás Bódai
In a low-order model of the general circulation of the atmosphere we examine the predictability of threshold exceedance events of certain observables. The likelihood of such binary events–the cornerstone also for the categoric (as opposed to probabilistic) prediction of threshold exceedences–is established from long time series of one or more observables of the same system. The prediction skill is measured by a summary index of the ROC curve that relates the hit- and false alarm rates. Our results for the examined systems suggest that exceedances of higher thresholds are more predictable; or in other words: rare large magnitude, i.e., extreme, events are more predictable than frequent typical events. We find this to hold provided that the bin size for binning time series data is optimized, but not necessarily otherwise. This can be viewed as a confirmation of a counterintuitive (and seemingly contrafactual) statement that was previously formulated for more simple autoregressive stochastic processes. However, we argue that for dynamical systems in general it may be typical only, but not universally true. We argue that when there is a sufficient amount of data depending on the precision of observation, the skill of a class of data-driven categoric predictions of threshold exceedences approximates the skill of the analogous model-driven prediction, assuming strictly no model errors. Therefore, stronger extremes in terms of higher threshold levels are more predictable both in case of data- and model-driven prediction. Furthermore, we show that a quantity commonly regarded as a measure of predictability, the finite-time maximal Lyapunov exponent, does not correspond directly to the ROC-based measure of prediction skill when they are viewed as functions of the prediction lead time and the threshold level. This points to the fact that even if the Lyapunov exponent as an intrinsic property of the system, measuring the instability of trajectories, determines predictability, it does that in a nontrivial manner.
PubDate: 2015-09-14T07:41:12Z
- Abstract: Publication date: Available online 10 September 2015
- Spectral transverse instabilities and soliton dynamics in the higher-order
multidimensional nonlinear Schrödinger equation- Abstract: Publication date: Available online 11 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Justin T. Cole, Ziad H. Musslimani
Spectral transverse instabilities of one-dimensional solitary wave solutions to the two-dimensional nonlinear Schrödinger (NLS) equation with fourth-order dispersion/diffraction subject to higher-dimensional perturbations are studied. A linear boundary value problem governing the evolution of the transverse perturbations is derived. The eigenvalues of the perturbations are numerically computed using Fourier and finite difference differentiation matrices. It is found that for both signs of the higher-order dispersion coefficient there exists a finite band of unstable transverse modes. In the long wavelength limit we derive an asymptotic formula for the perturbation growth rate that agrees well with the numerical findings. Using a variational formulation based on Lagrangian model reduction, an approximate expression for the perturbation eigenvalues is obtained and its validity is compared with both the asymptotic and numerical results. The time dynamics of a one-dimensional soliton stripe in the presence of a transverse perturbation is studied using direct numerical simulations. Numerical nonlinear stability analysis is also addressed.
PubDate: 2015-09-14T07:41:12Z
- Abstract: Publication date: Available online 11 September 2015
- Three-dimensional forced-damped dynamical systems with rich dynamics:
Bifurcations, chaos and unbounded solutions- Abstract: Publication date: Available online 9 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Tomoyuki Miyaji, Hisashi Okamoto, Alex D.D. Craik
We consider certain autonomous three-dimensional dynamical systems that can arise in mechanical and fluid-dynamical contexts. Extending a previous study in Craik and Okamoto (2002), to include linear forcing and damping, we find that the four-leaf structure discovered in that paper, and unbounded orbits, persist, but may now be accompanied by three distinct period-doubling cascades to chaos, and by orbits that approach stable equilibrium points. This rich structure is investigated both analytically and numerically, distinguishing three main cases determined by the damping and forcing parameter values.
PubDate: 2015-09-10T07:33:26Z
- Abstract: Publication date: Available online 9 September 2015
- Experiments on a non-smoothly-forced oscillator
- Abstract: Publication date: Available online 9 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Lawrence N. Virgin, Christopher George, Ashwath Kini
This paper describes some typical behavior encountered in the response of a harmonically-excited mechanical system in which a severe nonlinearity occurs due to an impact. Although such systems have received considerable recent attention (most of it from a theoretical viewpoint), the system scrutinized in this paper also involves a discrete input of energy at the impact condition. That is, it is kicked when contact is made. One of the motivations for this work is related to a classic pinball machine in which a ball striking a bumper experiences a sudden impulse, introducing additional unpredictability to the motion of the ball. A one-dimensional analog of a pinball machine was the subject of a detailed mathematical study in Pring and Budd (2011), and the current paper details behavior obtained from a mechanical experiment and describes dynamics not observed in a conventional (passive) impact oscillator.
PubDate: 2015-09-10T07:33:26Z
- Abstract: Publication date: Available online 9 September 2015
- Dispersive shock waves in nematic liquid crystals
- Abstract: Publication date: Available online 3 September 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Noel F. Smyth
The propagation of coherent light with an initial step intensity profile in a nematic liquid crystal is studied using modulation theory. The propagation of light in a nematic liquid crystal is governed by a coupled system consisting of a nonlinear Schrödinger equation for the light beam and an elliptic equation for the medium response. In general, the intensity step breaks up into a dispersive shock wave, or undular bore, and an expansion fan. In the experimental parameter regime for which the nematic response is highly nonlocal, this nematic bore is found to differ substantially from the standard defocusing nonlinear Schrödinger equation structure due to the effect of the nonlocality of the nematic medium. It is found that the undular bore is of Korteweg-de Vries equation-type, consisting of bright waves, rather than of nonlinear Schrödinger equation-type, consisting of dark waves. In addition, ahead of this Korteweg-de Vries bore there can be a uniform wavetrain with a short front which brings the solution down to the initial level ahead. It is found that this uniform wavetrain does not exist if the initial jump is below a critical value. Analytical solutions for the various parts of the nematic bore are found, with emphasis on the role of the nonlocality of the nematic medium in shaping this structure. Excellent agreement between full numerical solutions of the governing nematicon equations and these analytical solutions is found.
PubDate: 2015-09-05T18:05:47Z
- Abstract: Publication date: Available online 3 September 2015
- Editorial Board
- Abstract: Publication date: 1 August 2015
Source:Physica D: Nonlinear Phenomena, Volume 309
PubDate: 2015-08-31T19:27:50Z
- Abstract: Publication date: 1 August 2015
- Numerical and experimental observation of Arnol’d resonance webs in
an electrical circuit- Abstract: Publication date: Available online 31 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Naohiko Inaba, Kyohei Kamiyama, Takuji Kousaka, Tetsuro Endo
An extensive bifurcation analysis of partial and complete synchronizations of three-frequency quasi-periodic oscillations generated in an electric circuit is presented. Our model uses two-coupled hysteresis oscillators and a rectangular wave forcing term. The governing equation of the circuit is represented by a piecewise-constant dynamics generating a three-dimensional torus. The Lyapunov exponents are precisely calculated using explicit solutions without numerically solving any implicit equation. By analyzing this extremely simple circuit, we clearly demonstrate that it generates an extremely complex bifurcation structure called Arnol’d resonance web. Inevitably, chaos is observed in the neighborhood of Chenciner bubbles around which regions generating three-dimensional tori emanate. Furthermore, the numerical results are experimentally verified.
PubDate: 2015-08-31T19:27:50Z
- Abstract: Publication date: Available online 31 August 2015
- KAM tori and whiskered invariant tori for non-autonomous systems
- Abstract: Publication date: Available online 20 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Marta Canadell, Rafael de la Llave
We consider non-autonomous dynamical systems which converge to autonomous (or periodic) systems exponentially fast in time. Such systems appear naturally as models of many physical processes affected by external pulses. We introduce definitions of non-autonomous invariant tori and non-autonomous whiskered tori and their invariant manifolds and we prove their persistence under small perturbations, smooth dependence on parameters and several geometric properties (if the systems are Hamiltonian, the tori are Lagrangian manifolds). We note that such definitions are problematic for general time-dependent systems, but we show that they are unambiguous for systems converging exponentially fast to autonomous. The proof of persistence relies only on a standard implicit function theorem in Banach spaces and it does not require that the rotations in the tori are Diophantine nor that the systems we consider preserve any geometric structure. We only require that the autonomous system preserves these objects. In particular, when the autonomous system is integrable, we obtain the persistence of tori with rational rotational. We also discuss fast and efficient algorithms for their computation. The method also applies to infinite dimensional systems which define a good evolution, e.g. PDE’s. When the systems considered are Hamiltonian, we show that the time dependent invariant tori are isotropic. Hence, the invariant tori of maximal dimension are Lagrangian manifolds. We also obtain that the (un)stable manifolds of whiskered tori are Lagrangian manifolds. We also include a comparison with the more global theory developed in Blazevski and de la Llave (2011).
PubDate: 2015-08-23T09:51:06Z
- Abstract: Publication date: Available online 20 August 2015
- Stabilty of heteroclinic cycles in transverse bifurcations
- Abstract: Publication date: Available online 20 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Alexander Lohse
Heteroclinic cycles and networks exist robustly in dynamical systems with symmetry. They can be asymptotically stable, and gradually lose this stability through a variety of bifurcations, displaying different forms of non-asymptotic stability along the way. We analyse the stability change in a transverse bifurcation for different types of simple cycles in R 4 . This is done by first showing how stability of the cycle or network as a whole is related to stability indices along its connections — in particular, essential asymptotic stability is equivalent to all local stability indices being positive. Then we study the change of the stability indices. We find that all cycles of types B and C are generically essentially asymptotically stable after a transverse bifurcation, and that no type B cycle can be almost completely unstable (unlike type C cycles).
PubDate: 2015-08-23T09:51:06Z
- Abstract: Publication date: Available online 20 August 2015
- Simple analytic approximations for the Blasius problem
- Abstract: Publication date: Available online 13 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): R. Iacono, John P. Boyd
The classical boundary layer problem formulated by Heinrich Blasius more than a century ago is revisited, with the purpose of deriving simple and accurate analytical approximations to its solution. This is achieved through the combined use of a generalized Padé approach and of an integral iteration scheme deviced by Hermann Weyl. The iteration scheme is also used to derive very accurate bounds for the value of the second derivative of the Blasius function at the origin, which plays a crucial role in this problem.
PubDate: 2015-08-15T04:48:40Z
- Abstract: Publication date: Available online 13 August 2015
- Cellular non-deterministic automata and partial differential equations
- Abstract: Publication date: Available online 11 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): D. Kohler, J. Müller, U. Wever
We define cellular non-deterministic automata (CNDA) in the spirit of non-deterministic automata theory. They are different from the well-known stochastic automata. We propose the concept of deterministic superautomata to analyze the dynamical behavior of a CNDA and show especially that a CNDA can be embedded in a deterministic cellular automaton. As an application we discuss a connection between certain partial differential equations and CNDA.
PubDate: 2015-08-15T04:48:40Z
- Abstract: Publication date: Available online 11 August 2015
- Variational integrators for nonvariational partial differential equations
- Abstract: Publication date: Available online 13 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Michael Kraus, Omar Maj
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether’s theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa’s discretization of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.
PubDate: 2015-08-15T04:48:40Z
- Abstract: Publication date: Available online 13 August 2015
- Editorial Board
- Abstract: Publication date: 15 July 2015
Source:Physica D: Nonlinear Phenomena, Volume 308
PubDate: 2015-08-10T04:29:17Z
- Abstract: Publication date: 15 July 2015
- Blow up criterion of strong solution for 3D viscous liquid–gas
two-phase flow model with vacuum- Abstract: Publication date: 1 August 2015
Source:Physica D: Nonlinear Phenomena, Volume 309
Author(s): Lili Du, Qin Zhang
In this paper, we establish a blow-up criterion to the local strong solution to the three dimensional (3D) viscous liquid–gas two-phase flow model only in terms of the divergence of the velocity field. Moreover, the initial vacuum is allowed, and there is no extra restriction on viscous coefficients. Both the Cauchy problem and initial-boundary value problem are considered in this paper.
PubDate: 2015-08-10T04:29:17Z
- Abstract: Publication date: 1 August 2015
- Derivation of a wave kinetic equation from the resonant-averaged
stochastic NLS equation- Abstract: Publication date: 1 August 2015
Source:Physica D: Nonlinear Phenomena, Volume 309
Author(s): Sergei Kuksin, Alberto Maiocchi
We suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schrödinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analysed with full rigour in Kuksin and Maiocchi (0000), and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the Kolmogorov–Zakharov spectra.
PubDate: 2015-08-10T04:29:17Z
- Abstract: Publication date: 1 August 2015
- Equivariant Hopf bifurcation with general pressure laws
- Abstract: Publication date: Available online 6 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Tong Li, Jinghua Yao
The equivariant Hopf bifurcation dynamics of a class of system of partial differential equations is studied. The connections between the current dynamics and fundamental concepts in hyperbolic conservation laws are explained. The unique approximation property of center manifold reduction function is used in the current work to determine certain parameter in the normal form. The current work generalizes the study of the second author (J. Yao, 2014) and discovers a class of examples of O ( 2 ) Hopf bifurcation with two parameters arising from systems of partial differential equations.
PubDate: 2015-08-10T04:29:17Z
- Abstract: Publication date: Available online 6 August 2015
- The scattering transform for the Benjamin–Ono equation in the
small-dispersion limit- Abstract: Publication date: Available online 6 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Peter D. Miller, Alfredo N. Wetzel
Using exact formulae for the scattering data of the Benjamin–Ono equation valid for general rational potentials recently obtained in Miller and Wetzel (2015), we rigorously analyze the scattering data in the small-dispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself. Our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering, and they also provide new details not previously reported in the literature.
PubDate: 2015-08-10T04:29:17Z
- Abstract: Publication date: Available online 6 August 2015
- A computational study of residual KPP front speeds in time-periodic
cellular flows in the small diffusion limit- Abstract: Publication date: Available online 7 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Penghe Zu, Long Chen, Jack Xin
The minimal speeds ( c ∗ ) of the Kolmogorov-Petrovsky-Piskunov (KPP) fronts at small diffusion ( ϵ ≪ 1 ) in a class of time-periodic cellular flows with chaotic streamlines is investigated in this paper. The variational principle of c ∗ reduces the computation to that of a principle eigenvalue problem on a periodic domain of a linear advection-diffusion operator with space-time periodic coefficients and small diffusion. To solve the advection dominated time-dependent eigenvalue problem efficiently over large time, a combination of spectral methods and finite element, as well as the associated fast solvers, are utilized to accelerate computation. In contrast to the scaling c ∗ = O ( ϵ 1 / 4 ) in steady cellular flows, a new relation c ∗ = O ( 1 ) as ϵ ≪ 1 is revealed in the time-periodic cellular flows due to the presence of chaotic streamlines. Residual propagation speed emerges from the Lagrangian chaos which is quantified as a sub-diffusion process.
PubDate: 2015-08-10T04:29:17Z
- Abstract: Publication date: Available online 7 August 2015
- A geometric singular perturbation approach for planar stationary shock
waves- Abstract: Publication date: Available online 5 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Zhuopu Wang, Jiazhong Zhang, Junheng Ren, Muhammad Nauman Aslam
The non-linear non-equilibrium nature of shock waves in gas dynamics is investigated for the planar case. Along each streamline, the Euler equations with non-equilibrium pressure are reduced to a set of ordinary differential equations defining a slow-fast system, and geometric singular perturbation theory is applied. The proposed theory shows that an orbit on the slow manifold corresponds to the smooth part of the solution to the Euler equation, where non-equilibrium effects are negligible; and a relaxation motion from the unsteady to the steady branch of the slow manifold corresponds to a shock wave, where the flow relaxes from non-equilibrium to equilibrium. Recognizing the shock wave as a fast motion is found to be especially useful for shock wave detection when post-processing experimental measured or numerical calculated flow fields. Various existing shock detection methods can be derived from the proposed theory in a rigorous mathematical manner. The proposed theory provides a new shock detection method based on its non-linear non-equilibrium nature, and may also serve as the theoretical foundation for many popular shock wave detection techniques.
PubDate: 2015-08-06T04:10:21Z
- Abstract: Publication date: Available online 5 August 2015
- Averaging and spectral properties for the 2D advection-diffusion equation
in the semi-classical limit for vanishing diffusivity- Abstract: Publication date: Available online 3 August 2015
Source:Physica D: Nonlinear Phenomena
Author(s): J. Vukadinovic, E. Dedits, A.C. Poje, T. Schäfer
We consider the two-dimensional advection-diffusion equation on a bounded domain subject to Dirichlet or von Neumann boundary conditions involving a Liouville integrable Hamiltonian. Transformation to action-angle coordinates permits averaging in time and angle, resulting in an equation that allows for separation of variables. The Fourier transform in the angle coordinate transforms the equation into an effective diffusive equation and a countable family of non-self-adjoint Schrödinger equations. For the corresponding Liouville–Sturm problem, we apply complex-plane WKB methods to study the spectrum in the semi-classical limit for vanishing diffusivity. The spectral limit graph is found to consist of analytic curves (branches) related to Stokes graphs forming a tree-structure. Eigenvalues in the neighborhood of branches emanating from the imaginary axis are subject to various sublinear power laws with respect to diffusivity, leading to convection-enhanced rates of dissipation of the corresponding modes. The solution of ADE converges in the limit of vanishing diffusivity to the solution of the effective diffusion equation on convective time scales that are sublinear with respect to the diffusive time scales.
PubDate: 2015-08-06T04:10:21Z
- Abstract: Publication date: Available online 3 August 2015
- A computational overview of the solution space of the imaginary
Painlevé II equation- Abstract: Publication date: Available online 30 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Bengt Fornberg, J.A.C. Weideman
The six Painlevé equations were first formulated about a century ago. Since the 1970’s, it has become increasingly recognized that they play a fundamental role in a wide range of physical applications. A recently developed numerical pole field solver (Fornberg and Weideman, 2011) now allows their complete solutions spaces to be surveyed across the complex plane. Following such surveys of the P I , P I I and P I V equations, we consider here the case of the imaginary P I I equation (the standard P I I equation, with a change of sign for its nonlinear term). Solutions to this equation share many features with other classes of Painlevé transcendents, including a rich variety of pole field configurations, with connection formulas linking asymptotic behaviors in different directions of the complex plane.
Graphical abstract
PubDate: 2015-08-02T03:25:09Z
- Abstract: Publication date: Available online 30 July 2015
- Maxwell’s conjecture on three point charges with equal magnitudes
- Abstract: Publication date: Available online 30 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Ya-Lun Tsai
Maxwell’s conjecture on three point charges states that the number of non-degenerate equilibrium points of the electrostatic field generated by them in R 3 is at most four. We prove the conjecture in the cases when three point charges have equal magnitudes and show the number of isolated equilibrium points can only be zero, two, three, or four. Specifically, fixing positions of two positive charges in R 3 , we know exactly where to place the third positive charge to have two, three, or four equilibrium points. All equilibrium points are isolated and there are no other possibilities for the number of isolated equilibrium points. On the other hand, if both two of the fixed charges have negative charge values, there are always two equilibrium points except when the third positive charge lies in the line segment connecting the two negative charges. The exception cases are when the field contains only a curve of equilibrium points. In this paper, computations assisted by computer involve symbolic and exact integer computations. Therefore, all the results are proved rigorously.
PubDate: 2015-08-02T03:25:09Z
- Abstract: Publication date: Available online 30 July 2015
- The Whitham Equation as a model for surface water waves
- Abstract: Publication date: Available online 31 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Daulet Moldabayev, Henrik Kalisch, Denys Dutykh
The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to different free surface models: the KdV equation, the BBM equation, and the Padé (2,2) model. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than the three considered models.
PubDate: 2015-08-02T03:25:09Z
- Abstract: Publication date: Available online 31 July 2015
- Global modes in nonlinear non-normal evolutionary models: Exact solutions,
perturbation theory, direct numerical simulation, and chaos- Abstract: Publication date: Available online 26 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): LennonÓ. Náraigh
This paper is concerned with the theory of generic non-normal nonlinear evolutionary equations, with potential applications in Fluid Dynamics and Optics. Two theoretical models are presented. The first is a model two-level non-normal nonlinear system that not only highlights the phenomena of linear transient growth, subcritical transition and global modes, but is also of potential interest in its own right in the field of nonlinear optics. The second is the fairly familiar inhomogeneous nonlinear complex Ginzburg–Landau (CGL) equation. The two-level model is exactly solvable for the nonlinear global mode and its stability, while for the spatially-extended CGL equation, perturbative solutions for the global mode and its stability are presented, valid for inhomogeneities with arbitrary scales of spatial variation and global modes of small amplitude, corresponding to a scenario near criticality. For other scenarios, a numerical iterative nonlinear eigenvalue technique is preferred. Two global modes of different amplitudes are revealed in the numerical approach. For both the two-level system and the nonlinear CGL equation, the analytical calculations are supplemented with direct numerical simulation, thus showing the fate of unstable global modes. For the two-level model this results in unbounded growth of the full nonlinear equations. For the spatially-extended CGL model in the subcritical regime, the global mode of larger amplitude exhibits a ‘one-sided’ instability leading to a chaotic dynamics, while the global mode of smaller amplitude is always unstable (theory confirms this). However, advection can stabilize the mode of larger amplitude.
PubDate: 2015-07-28T20:53:41Z
- Abstract: Publication date: Available online 26 July 2015
- Detecting changes in coupling with Granger causality method from time
series with fast transient processes- Abstract: Publication date: Available online 26 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Ilya V. Sysoev, Marina V. Sysoeva
The ability of the Granger causality method to detect directed coupling between subsystems of a complex system in a moving time window is investigated on etalon oscillators. In particular, the time series consisting of alternate stationary regimes characterised by the different amplitude and shape of oscillations with fast transient processes between these regimes are considered, with similar transitions being possible due to changes either in the coupling or in the individual properties of subsystems. Two popular approaches to surrogate times series generation are used to check the significance of the method results. Two model structures: the standard linear and the special non-linear adapted to data are implemented. The Granger causality method using the model structure adapted to data is shown to be significantly advantageous in detecting coupling directionality and the instant time of the regime change than the standard linear method, while in some cases the sensitivity and the specificity of the adapted approach are insufficient.
PubDate: 2015-07-28T20:53:41Z
- Abstract: Publication date: Available online 26 July 2015
- Low-frequency variability and heat transport in a low-order nonlinear
coupled ocean-atmosphere model- Abstract: Publication date: Available online 28 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Stéphane Vannitsem, Jonathan Demaeyer, Lesley De Cruz, Michael Ghil
We formulate and study a low-order nonlinear coupled ocean–atmosphere model with an emphasis on the impact of radiative and heat fluxes and of the frictional coupling between the two components. This model version extends a previous 24-variable version by adding a dynamical equation for the passive advection of temperature in the ocean, together with an energy balance model. The bifurcation analysis and the numerical integration of the model reveal the presence of low-frequency variability (LFV) concentrated on and near a long-periodic, attracting orbit. This orbit combines atmospheric and oceanic modes, and it arises for large values of the meridional gradient of radiative input and of frictional coupling. Chaotic behavior develops around this orbit as it loses its stability; this behavior is still dominated by the LFV on decadal and multi-decadal time scales that is typical of oceanic processes. Atmospheric diagnostics also reveals the presence of predominant low- and high-pressure zones, as well as of a subtropical jet; these features recall realistic climatological properties of the oceanic atmosphere. Finally, a predictability analysis is performed. Once the decadal-scale periodic orbits develop, the coupled system’s short-term instabilities—as measured by its Lyapunov exponents—are drastically reduced, indicating the ocean’s stabilizing role on the atmospheric dynamics. On decadal time scales, the recurrence of the solution in a certain region of the invariant subspace associated with slow modes displays some extended predictability, as reflected by the oscillatory behavior of the error for the atmospheric variables at long lead times.
PubDate: 2015-07-28T20:53:41Z
- Abstract: Publication date: Available online 28 July 2015
- Dynamics and statistics of noise-like pulses in modelocked lasers
- Abstract: Publication date: Available online 21 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Graham M. Donovan
Noise-like pulses and optical rogue waves are connected nonlinear phenomena which can occur in passively modelocked laser systems. Here we consider a range of model systems to explore the conditions under which noise-like pulses can be expected to occur, and further when the resulting statistics meet the optical rogue wave criteria. We show, via a series of careful simulations, that noise-like pulses and optical rogue waves can arise either separately or together, and that they may emerge from standard soliton-like solutions via different mechanisms. We also propose a quantitative definition of noise-like pulses, and explore the issues in carefully convergence testing numerical methods for such systems.
PubDate: 2015-07-25T10:32:40Z
- Abstract: Publication date: Available online 21 July 2015
- Bandcount adding structure and collapse of chaotic attractors in a
piecewise linear bimodal map- Abstract: Publication date: Available online 17 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Viktor Avrutin, Manuel Clüver, Vincent Mahout, Danièle Fournier-Prunaret
In this work we investigate bifurcation structures in the chaotic domain of a piecewise linear bimodal map. The map represents a model of a circuit proposed to generate chaotic signals. For practical purposes it is necessary that the map generates robust broad-band chaos. However, experiments show that this requirement is fulfilled not everywhere. We show that the chaotic domain in the parameter space of this map contains regions in which the map has multi-band chaotic attractors. These regions are confined by bifurcation curves associated with homoclinic bifurcations of unstable cycles, and form a so-called bandcount adding structure previously reported to occur in discontinuous maps. Additionally, it is shown that inside each of these regions chaotic attractors collapse to particular cycles existing on a domain of zero measure in the parameter space and organized in a period adding structure in the form known for circle maps.
PubDate: 2015-07-21T10:27:24Z
- Abstract: Publication date: Available online 17 July 2015
- Thermoconvective instabilities to explain the main characteristics of a
dust devil-like vortex- Abstract: Publication date: Available online 10 July 2015
Source:Physica D: Nonlinear Phenomena
Author(s): M.C. Navarro , D. Castaño , H. Herrero
In this paper we show numerically that the main characteristics of a dust devil-like vortex: vertical vorticity generation, eye formation, and tilting of the eye/axis of rotation, can be explained by thermoconvective mechanisms. By considering a cylinder non-homogeneously heated from below we prove that an intense localized heating on the ground generates a convective stationary axisymmetric flow that begins to spiral up around a central axis when perturbation vertical vorticity is permitted and a critical vertical temperature gradient is exceeded, thus forming an axisymmetric vortex. If the intense heating on the ground is not too localized and the temperature gradient continues increasing, central downdrafts appear in the vortex and an eye is formed. We show that the axisymmetric vortex loses stability toward a new state for which the axisymmetry is broken, the axis of rotation or proper eye displaces from the center and tilts. The vortical states found are comparable to dust devils. These findings establish the relevance of thermoconvection on the formation and evolution of these atmospheric phenomena.
PubDate: 2015-07-11T02:58:12Z
- Abstract: Publication date: Available online 10 July 2015
- O(2) Hopf bifurcation of viscous shock waves in a channel
- Abstract: Publication date: 15 July 2015
Source:Physica D: Nonlinear Phenomena, Volume 308
Author(s): Alin Pogan , Jinghua Yao , Kevin Zumbrun
Extending work of Texier and Zumbrun in the semilinear non-reflection symmetric case, we study O ( 2 ) transverse Hopf bifurcation, or “cellular instability”, of viscous shock waves in a channel, for a class of quasilinear hyperbolic–parabolic systems including the equations of thermoviscoelasticity. The main difficulties are to (i) obtain Fréchet differentiability of the time- T solution operator by appropriate hyperbolic–parabolic energy estimates, and (ii) handle O ( 2 ) symmetry in the absence of either center manifold reduction (due to lack of spectral gap) or (due to nonstandard quasilinear hyperbolic–parabolic form) the requisite framework for treatment by spatial dynamics on the space of time-periodic functions, the two standard treatments for this problem. The latter issue is resolved by Lyapunov–Schmidt reduction of the time- T map, yielding a four-dimensional problem with O ( 2 ) plus approximate S 1 symmetry, which we treat “by hand” using direct Implicit Function Theorem arguments. The former is treated by balancing information obtained in Lagrangian coordinates with that from associated constraints. Interestingly, this argument does not apply to gas dynamics or magnetohydrodynamics (MHD), due to the infinite-dimensional family of Lagrangian symmetries corresponding to invariance under arbitrary volume-preserving diffeomorphisms.
PubDate: 2015-07-11T02:58:12Z
- Abstract: Publication date: 15 July 2015
- Domain coarsening in a subdiffusive Allen–Cahn equation
- Abstract: Publication date: Available online 26 June 2015
Source:Physica D: Nonlinear Phenomena
Author(s): M. Abu Hamed , A.A. Nepomnyashchy
Domain coarsening in a one-dimensional bistable system governing by a subdiffusive generalization of the Allen–Cahn equation is considered. Integro-differential equations governing the motion of interacting domain walls are derived and solved analytically and numerically. The dependence of the domain wall dynamics on the subdiffusion parameter is investigated.
PubDate: 2015-07-01T11:33:57Z
- Abstract: Publication date: Available online 26 June 2015
- A multidomain model for ionic electrodiffusion and osmosis with an
application to cortical spreading depression- Abstract: Publication date: Available online 29 June 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Yoichiro Mori
Ionic electrodiffusion and osmotic water flow are central processes in many physiological systems. We formulate a system of partial differential equations that governs ion movement and water flow in biological tissue. A salient feature of this model is that it satisfies a free energy identity, ensuring the thermodynamic consistency of the model. A numerical scheme is developed for the model in one spatial dimension and is applied to a model of cortical spreading depression, a propagating breakdown of ionic and cell volume homeostasis in the brain.
PubDate: 2015-07-01T11:33:57Z
- Abstract: Publication date: Available online 29 June 2015
- Traveling wave profiles for a crystalline front invading liquid states:
Analytical and numerical solutions- Abstract: Publication date: 15 July 2015
Source:Physica D: Nonlinear Phenomena, Volume 308
Author(s): P.K. Galenko , F. Iunes Sanches , K.R. Elder
The properties of a two dimensional crystalline phase invading a metastable or unstable liquid state are examined using the amplitude expansion formulation of the hyperbolic and parabolic phase-field crystal model. When the amplitudes are real and equal to each other, analytic expressions are derived for the profile of a steady state liquid–solid front traveling at constant velocity. Numerical simulations of the full amplitude formulation are conducted and compared with the analytic results. Close to the melting transition the analytic results for the liquid–solid profile, velocity and width are in quantitative agreement with the numerical results and disagree far from the transition.
PubDate: 2015-06-26T14:28:59Z
- Abstract: Publication date: 15 July 2015
- Phase field based nonlocal anisotropic damage mechanics model
- Abstract: Publication date: 15 July 2015
Source:Physica D: Nonlinear Phenomena, Volume 308
Author(s): Navid Mozaffari , George Z. Voyiadjis
A nonlocal anisotropic damage theory is developed in this work through the phase field method to address the anisotropic damage evolution in materials. The anisotropic damage is discussed and appropriate nonconserved order parameters in three mutually perpendicular directions are defined to find the growth of the components of a second order diagonal damage tensor corresponding to the principal directions of a general second order damage tensor. In contrast to the previous models, two new tensors are proposed to act as interpolation and potential functions along with the Allen–Cahn equation in order to obtain the evolution of the order parameters, which is the basis of the definition of the damage rate. The tensor formulation of the growth of the components of the damage tensor is proposed for the first time. It is shown that, by introducing a set of material parameters including a length scale parameter due to damage, there is a robust and simplified way to model the nonlocal behavior of damage and predict the corresponding material behavior as components of a second order diagonal damage tensor.
PubDate: 2015-06-26T14:28:59Z
- Abstract: Publication date: 15 July 2015
- Dynamical Hamiltonian–Hopf instabilities of periodic traveling waves
in Klein–Gordon equations- Abstract: Publication date: Available online 25 June 2015
Source:Physica D: Nonlinear Phenomena
Author(s): R. Marangell , P.D. Miller
We study the unstable spectrum close to the imaginary axis for the linearization of the nonlinear Klein–Gordon equation about a periodic traveling wave in a co-moving frame. We define dynamical Hamiltonian–Hopf instabilities as points in the stable spectrum that are accumulation points for unstable spectrum, and show how they can be determined from the knowledge of the discriminant of an associated Hill’s equation. This result allows us to give simple criteria for the existence of dynamical Hamiltonian–Hopf instabilities in terms of instability indices previously shown to be useful in stability analysis of periodic traveling waves. We also discuss how these methods can be applied to more general nonlinear wave equations.
PubDate: 2015-06-26T14:28:59Z
- Abstract: Publication date: Available online 25 June 2015
- On Slater’s criterion for the breakup of invariant curves
- Abstract: Publication date: Available online 24 June 2015
Source:Physica D: Nonlinear Phenomena
Author(s): C.V. Abud , I.L. Caldas
We numerically explore Slater’s theorem in the context of dynamical systems to study the breakup of invariant curves. Slater’s theorem states that an irrational translation over a circle returns to an arbitrary interval in at most three different recurrence times expressible by the continued fraction expansion of the related irrational number. The hypothesis considered in this paper is that Slater’s theorem can be also verified in the dynamics of invariant curves. Hence, we use Slater’s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of area-preserving maps.
PubDate: 2015-06-26T14:28:59Z
- Abstract: Publication date: Available online 24 June 2015
- Extreme phase sensitivity in systems with fractal isochrons
- Abstract: Publication date: Available online 19 June 2015
Source:Physica D: Nonlinear Phenomena
Author(s): A. Mauroy , I. Mezić
Sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors. However, even systems with (quasi)periodic dynamics can exhibit it. In this context we report on the fractal properties of the isochrons of some continuous-time asymptotically periodic systems. We define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons. Similar results are also obtained with discrete-time systems. As an illustration of the framework, we compute the phase sensitivity coefficient for popular models of bursting neurons, suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive (unreliable) phase response.
PubDate: 2015-06-26T14:28:59Z
- Abstract: Publication date: Available online 19 June 2015
- Transport bounds for a truncated model of Rayleigh–Bénard
convection- Abstract: Publication date: Available online 10 June 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Andre N. Souza , Charles R. Doering
We investigate absolute limits on heat transport in a truncated model of Rayleigh–Bénard convection. Two complementary mathematical approaches—a background method analysis and an optimal control formulation—are used to derive upper bounds in a distinguished eight-ODE model proposed by Gluhovsky, Tong, and Agee. In the optimal control approach the flow no longer obeys an equation of motion, but is instead a control variable. Both methods produce the same estimate, but in contrast to the analogous result for the seminal three-ODE Lorenz system, the best upper bound apparently does not always correspond to an exact solution of the equations of motion.
PubDate: 2015-06-26T14:28:59Z
- Abstract: Publication date: Available online 10 June 2015
- Nonlinear conductance and heterogeneity of voltage-gated ion channels
allow defining electrical surface domains in cell membranes- Abstract: Publication date: Available online 6 June 2015
Source:Physica D: Nonlinear Phenomena
Author(s): Javier Cervera , José A. Manzanares , Salvador Mafe
The membrane potential of a cell measured by typical electrophysiological methods is only an average magnitude and experimental techniques allowing a more detailed mapping of the cell surface have shown the existence of spatial domains with locally different electric potentials and currents. Electrical potentials in non-neural cells are regulated by the nonlinear conductance of membrane ion channels. Voltage-gated potassium channels participate in cell hyperpolarization/depolarization processes and control the electrical signals over the cell surface, constituting good candidates to study basic biological questions on a more simplified scale than the complex cell membrane. These channels show also a high heterogeneity, making it possible to analyze the effects of diversity in the electrical responses of channels localized on spatial domains. We use a phenomenological approach of voltage gating that reproduces the observed rectification characteristics of inward rectifying potassium channels and relate the threshold voltage heterogeneity of the channels to the establishment of spatial domains with different electrical sensitivities. Although our model is only a limited picture of the whole cell membrane, it shows that domains with different ion channels may permit or suppress steady state bioelectrical signals over the cell surface according to their particular voltage sensitivity. Also, the nonlinear electrical coupling of channels with different threshold potentials can lead to a rich variety of bioelectrical phenomena, including regions of membrane potential bi-stability.
Graphical abstract
PubDate: 2015-06-26T14:28:59Z
- Abstract: Publication date: Available online 6 June 2015